What are four different measures of the amount of water vapor in the air?
(In addition to naming them, you should know how each is defined.)

Mixing Ratio (a direct measure of humidity because it is the number of grams of water vapor prsent in a kilogram of dry air. You can get this from a sounding diagram)

Relative Humidity (an indirect measure humidity, and a way of deducing how "close" the parcel is to saturation, and is a ratio of mixing ratio to the saturation mixing ratio. Synoptic meteorologists use relative humidity as a measure of how "close" the atmosphere is to producing a cloud or condensation at the observing site. Unless the saturation mixing ratio is known, however, relative humidity does not directly inform you how much water vapor is present)

Dewpoint Temperature (the temperature to which an air parcel at a given pressure must be cooled in order to reduce the saturation mixing ratio [amount of water vapor the kilogram air pacel could hold] to the value of the mixing ratio [amount of water vapor actually present in the kilogram air parcel]. It can easily be shown that at a given pressure, the dew point temperature is directly proportional to the mixing ratio, and the value of the mixing ratio can be obtained by plotting the dew point temperature on an atmospheric sounding diagram)

Wet-bulb Temperature i(s measured using a standard mercury-in-glass thermometer, with the thermometer bulb wrapped in muslin, which is kept wet. The evaporation of water from the thermometer has a cooling effect, so the temperature indicated by the wet bulb thermometer is less than the temperature indicated by a dry-bulb (normal, unmodified) thermometer. The rate of evaporation from the wet-bulb thermometer depends on the humidity of the air - evaporation is slower when the air is already full of water vapour. For this reason, the difference in the temperatures indicated by the two thermometers gives a measure of atmospheric humidity).

What does it mean when we say that a "parcel" of air is saturated
with water vapor?

Saturation: With respect to a plane surface of wate surmounted by a vacuumr, when the number of water vapor molecules leaving the liquid is balanced by the number returning, the equilibrium state is known as "saturation". The number of water vapor molecules leaving the liquid is dependent only upon the temperature of the liquid, and the water vapor molecules "filling" in the vacuum will have the same temperature. The pressure at which the equilibrium occurs is called saturation vapor pressure and the mixing ratio at this equilibrium is termed saturation mixing ratio.

There is an upper limit to the amount of water that can be present
in a parcel of air. (You can think of this upper limit as the maximum capacity
of the parcel to "hold" water vapor, though this terminology can
be misleading.) What is one measure of the upper limit to the amount of water
vapor that can be present in a parcel of air? What property of the air determines
what this upper limit is?

The pressure at which the equilibrium occurs is called saturation vapor pressure and the mixing ratio at this equilibrium is termed saturation mixing ratio. The first is a measure of the amount of water that can be present at maxium in a parcel of air as determined by the pressure that vapor would exert, and the second is a measure of the amoutn of water that can be present at maxium as determined by the number of grams of water vapor per kilogram of dry air.

Suppose that a parcel of air is not in contact with liquid water. What
happens to the amount of water vapor in the parcel (expressed in terms of
its mixing ratio) if the parcel warms up? Why?

In this case, since there is no additional source of water vapor, nor is there a surface upon which the molecules to condense (leave the parcel), then the amount of water vapor is independent of the temperature of the air parcel.

What has to happen for the amount of water vapor in the parcel (expressed
in terms of its mixing ratio) to increase?

A source of water has to be encountered by the air parcel. Then, initially, more water vapor molecles will escape the liquid and change the mixing ratio.

What happens to the maximum capacity of a parcel of air to "hold" water
vapor (expressed in terms of its saturation vapor pressure or saturation mixing
ratio) when the parcel warms up? When it cools? What happens to the maximum
capacity of the parcel to hold water vapor (expressed in terms of its saturation
vapor pressure or saturation mixing ratio) when liquid water evaporates into
the air (assuming the parcel's temperature and total pressure don't change)?

When the parcel warms up, the kinetic energy of the water vapor molecules increases. The chances of individual molecules returning to the liquid decreases, and the equilibrium will occur at a higher vapor pressure or mixing ratio. This is because the number of molecules leaving the liquid will now not be balanced by the number returning. The number entering the air parcel will increase until the random collisions on the liquid surface balances the input and outputs again. Vice versa.

When the amount of water vapor in the air parcel increases, supersaturation will occur. In that case condesnation will exceeld evaporation, causing the saturation vapor pressure to decrease.

What happens to the relative humidity of an air parcel when the parcel warms
up? When it cools off (assuming that parcel starts and stays unsaturated)?
What happens to the relative humidity in the parcel when liquid water evaporates
into it (assuming that its temperature doesn't change)?

Relative humidity decreases when the parcel warms up. Relative humidity increases when the air parcel cools down. When liquid water evaporates intot he air parcel, the mixing ratio increases relative to the saturation mixing ratio, hence the relative humidity increases.

What is the relative humidity of air that is saturated with water vapor?

The relative humidity of air that is saturated with water vapor is 100%.

Consider two parcels of air that have equal amounts of air in them, not
counting water vapor—that is, the dry parts of each parcel have equal
mass.) One of the parcel has a temperature of 68°F (20°C, room temperature)
and is saturated. The other parcel is also saturated but its temperature is
32°F (0°C)? Which parcel has more water vapor in it? Why?

The cooler air parcel has more water vapor in it. This is because though the relative humidity is 100% for both, the saturation mixing ratio for the cooler air parcel has a much smaller value. In general, increasing temperature by 10C doubles the saturation mixing ratio.

Which has more water vapor in it: warm air or cool air? (Assume two parcels
of air, the dry portions of which have equal mass.)

This question cannot be answered unless the mixing ratios are given. The saturation mixing ratio will be greater for the warmer air, though.

What happens to the amount of water vapor in air (expressed in terms of
the mixing ratio) when the air is cooled to its dew point? What happens when
it's cooled below it's dew point?

In the former case, nothing happens to the amount of water vapor, until condensation occurs. Below the dewpoint temperature, the amount of water vapor preent in excess of the saturation mixing ratio will begin to condense.

What happens to the relative humidity of saturated air that is cooled below
its dew point (at least, after a very brief period of transition)?

The relative humidity will continue to be 100%. The mixing ratio and saturation mixing ratios will decrease at the same rate, and 100% humidity will occur at a lower saturation mixing ratio.

What are two totally different ways to increase the relative humidity in
a very dry, heated apartment on a cold, dry winter day?

The relative humidity inside of a cloud has to be near or at 100% because condensation is occurring to produce the cloud droplets.

On a warm, humid day, if you pour cold lemonade from the refrigerator into
a glass, liquid water soon appears on the outside of the previously dry glass.
How does this happen?

The surface of the glass has been cooled by conduction. The air in contact with the chilled glass is chilled by conduction. The saturation mixing ratio of the chilled air is reduced. When the reduction is such that the saturation mixing ratio is the same as the mixing ratio, condensation occurs. This takes place on the chilled surface, resulting in dew formation on the glass surface.

Consider two parcels of air, one in Tucson, AZ and the other in San Francisco,
CA, each containing 1 kg of dry air plus some water vapor. Assume that they
are both at a pressure of about 1000 mb.

The parcel in Tucson has a relative humidity of 20% and a temperature
of 37°C (about 98.6°F). At that temperature, the parcel of air
is capable of "holding" at most about 45 grams of water vapor.

The parcel in San Francisco has a relative humidity of 50% and a temperature
of 20°C (68°F). At that temperature, this parcel of air is capable
of "holding" at most about 15 grams of water vapor.

Which parcel has the most water vapor in it?

Tucson: 0.20 X 45 g = 9 g; San Francisco, 0.50 X 15 g = 7.5 g. Despite the higher relative humidity at San Francisco, there is much more water vapor present in Tucson. From Fig. 4.10, Tucson's dew point would be 9.5C or so. San Francisco's dew point would be 0C.

What would happen to the amount of water vapor in the Tucson parcel if
it were cooled to the same temperature as the air in San Francisco? What
would its relative humidity be then?

The amount of water vapor would remain the same, however the relative humidity would increase, since the new saturation mixing ratio would be 15 g. The relative humidity would increase from 20% to 9/15 X 100 = 60%.